On balanced sequences and their critical exponent

TL;DR

This paper investigates aperiodic balanced sequences over finite alphabets, characterizing their structure via Sturmian sequences and constant gap sequences, and develops a method to compute their critical exponents, confirming a conjecture about minimal critical exponents.

Contribution

It introduces a new method for computing the critical exponent of balanced sequences with quadratic Sturmian slopes, linking their structure to return words and confirming a conjecture on minimal critical exponents.

Findings

The language of balanced sequences is eventually dendric.

A method for calculating critical exponents based on return words is developed.

Confirmed the minimal critical exponent conjecture for sequences over 9-letter and 0-letter alphabets.

Abstract

We study aperiodic balanced sequences over finite alphabets. A sequence vv of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y'. We show that the language of v is eventually dendric and we focus on return words to its factors. We develop a method for computing the critical exponent and asymptotic critical exponent of balanced sequences, provided the associated Sturmian sequence u has a quadratic slope. The method is based on looking for the shortest return words to bispecial factors in v. We illustrate our method on several examples; in particular we confirm a conjecture of Rampersad, Shallit and Vandomme that two specific sequences have the least critical exponent among all balanced sequences over 9-letter (resp., $0-letter) alphabets.